Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
For 2011, medium risk data is used in the high risk data set, as no
high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from
2007 to 2023. For 2007 to 2011 (both included) no high risk data is
available.
Summary of gross returns
## vmr pmr mmr vhr
## Min. :0.868 Min. :0.904 Min. :0.988 Min. :0.849
## 1st Qu.:1.044 1st Qu.:1.042 1st Qu.:1.013 1st Qu.:1.039
## Median :1.097 Median :1.084 Median :1.085 Median :1.099
## Mean :1.070 Mean :1.065 Mean :1.066 Mean :1.085
## 3rd Qu.:1.136 3rd Qu.:1.107 3rd Qu.:1.101 3rd Qu.:1.160
## Max. :1.168 Max. :1.141 Max. :1.133 Max. :1.214
## phr mhr
## Min. :0.878 Min. :0.977
## 1st Qu.:1.068 1st Qu.:1.013
## Median :1.128 Median :1.113
## Mean :1.095 Mean :1.087
## 3rd Qu.:1.182 3rd Qu.:1.128
## Max. :1.208 Max. :1.207
## vmrl
## Min. :0.801
## 1st Qu.:1.013
## Median :1.085
## Mean :1.061
## 3rd Qu.:1.128
## Max. :1.193
## vmr pmr mmr vhr phr mhr
## Min. : 0.868 0.904 0.988 0.849 0.878 0.977
## 1st Qu.: 1.044 1.042 1.013 1.039 1.068 1.013
## Median : 1.097 1.084 1.085 1.099 1.128 1.113
## Mean : 1.070 1.065 1.066 1.085 1.095 1.087
## 3rd Qu.: 1.136 1.107 1.101 1.160 1.182 1.128
## Max. : 1.168 1.141 1.133 1.214 1.208 1.207
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.988 | mmr | 1.068 | phr | 1.128 | phr | 1.095 | phr | 1.136 | vmr | 1.168 | vmr |
| 0.977 | mhr | 1.044 | vmr | 1.113 | mhr | 1.087 | mhr | 1.107 | pmr | 1.141 | pmr |
| 0.904 | pmr | 1.042 | pmr | 1.099 | vhr | 1.085 | vhr | 1.101 | mmr | 1.133 | mmr |
| 0.878 | phr | 1.039 | vhr | 1.097 | vmr | 1.070 | vmr | 1.160 | vhr | 1.214 | vhr |
| 0.868 | vmr | 1.013 | mmr | 1.085 | mmr | 1.066 | mmr | 1.182 | phr | 1.208 | phr |
| 0.849 | vhr | 1.013 | mhr | 1.084 | pmr | 1.065 | pmr | 1.128 | mhr | 1.207 | mhr |
## cov(vmr, pmr) = -0.001094875
## cov(vhr, phr) = -0.0001730651
##
## AIC: -27.8497
## BIC: -25.58991
## m: 0.0480931
## s: 0.1198426
## nu (df): 3.303595
## xi: 0.03361192
## R^2: 0.993
##
## An R^2 of 0.993 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 7.4 percent
## What is the risk of losing max 25 %? =< 1.8 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 41 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 278.437 kr.
## SD of portfolio index value after 20 years: 122.559 kr.
## Min total portfolio index value after 20 years: 1.432 kr.
## Max total portfolio index value after 20 years: 960.495 kr.
##
## Share of paths finishing below 100: 5.16 percent
##
## AIC: -34.35752
## BIC: -31.02467
## m: 0.05171176
## s: 0.1149408
## nu (df): 2.706099
## xi: 0.5049945
## R^2: 0.978
##
## An R^2 of 0.978 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 5.4 percent
## What is the risk of losing max 25 %? =< 1.3 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 36.2 percent
## What is the chance of gaining min 25 %? >= 0.3 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 292.991 kr.
## SD of portfolio index value after 20 years: 116.159 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1003.923 kr.
##
## Share of paths finishing below 100: 3.18 percent
##
## AIC: -21.42488
## BIC: -19.16508
## m: 0.06471454
## s: 0.1499924
## nu (df): 3.144355
## xi: 0.002367034
## R^2: 0.991
##
## An R^2 of 0.991 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 8.3 percent
## What is the risk of losing max 25 %? =< 2.5 percent
## What is the risk of losing max 50 %? =< 0.4 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 53.3 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 405.817 kr.
## SD of portfolio index value after 20 years: 217.397 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1483.003 kr.
##
## Share of paths finishing below 100: 3.86 percent
##
## AIC: -33.22998
## BIC: -30.97018
## m: 0.05789224
## s: 0.1234592
## nu (df): 2.265273
## xi: 0.477324
## R^2: 0.991
##
## An R^2 of 0.991 suggests that the fit is extremely good.
##
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.9 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 32.7 percent
## What is the chance of gaining min 25 %? >= 0.1 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.
## [1] -0.091256521 -0.003731241 0.027312079 0.045808232 0.059068633
## [6] 0.069575113 0.078454727 0.086316936 0.093536451 0.100370932
## [11] 0.107018607 0.114081432 0.127604387
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 319.768 kr.
## SD of portfolio index value after 20 years: 104.054 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 999.664 kr.
##
## Share of paths finishing below 100: 2.07 percent
##
## AIC: -23.72565
## BIC: -21.46585
## m: 0.08386034
## s: 0.1210107
## nu (df): 3.184569
## xi: 0.01790306
## R^2: 0.964
##
## An R^2 of 0.964 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 5.3 percent
## What is the risk of losing max 25 %? =< 1.4 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 59.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 555.977 kr.
## SD of portfolio index value after 20 years: 244.76 kr.
## Min total portfolio index value after 20 years: 0.626 kr.
## Max total portfolio index value after 20 years: 1748.121 kr.
##
## Share of paths finishing below 100: 0.79 percent
##
## AIC: -36.9603
## BIC: -34.7005
## m: 0.05902873
## s: 0.08757749
## nu (df): 2.772621
## xi: 0.02904471
## R^2: 0.89
##
## An R^2 of 0.89 suggests that the fit is not completely random.
##
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.7 percent
## What is the risk of losing max 50 %? =< 0.1 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 35.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The fit suggests big losses for the lowest percentiles, which are not
present in the data.
So the fit is actually a very cautious estimate.
Let’s plot the fit and the observed returns together.
Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.
## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
##
## Mean portfolio index value after 20 years: 324.319 kr.
## SD of portfolio index value after 20 years: 97.388 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 673.26 kr.
##
## Share of paths finishing below 100: 0.97 percent
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 301.885 kr.
## SD of portfolio index value after 20 years: 82.893 kr.
## Min total portfolio index value after 20 years: 18.971 kr.
## Max total portfolio index value after 20 years: 1997.325 kr.
##
## Share of paths finishing below 100: 0.33 percent
##
## AIC: -24.26084
## BIC: -22.00104
## m: 0.0822419
## s: 0.07129843
## nu (df): 89.86289
## xi: 0.7697502
## R^2: 0.961
##
## An R^2 of 0.961 suggests that the fit is very good.
##
## What is the risk of losing max 10 %? =< 0.9 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
##
## What is the chance of gaining min 10 %? >= 46.1 percent
## What is the chance of gaining min 25 %? >= 1.2 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent
The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
We see that the high risk mix provides a much better upside and smaller downside.
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 501.245 kr.
## SD of portfolio index value after 20 years: 158.271 kr.
## Min total portfolio index value after 20 years: 129.151 kr.
## Max total portfolio index value after 20 years: 1335.386 kr.
##
## Share of paths finishing below 100: 0 percent
## Down-and-out simulation:
## Probability of down-and-out: 0 percent
##
## Mean portfolio index value after 20 years: 475.563 kr.
## SD of portfolio index value after 20 years: 163.327 kr.
## Min total portfolio index value after 20 years: 25.692 kr.
## Max total portfolio index value after 20 years: 1204.922 kr.
##
## Share of paths finishing below 100: 0.15 percent
Risk of max loss of x percent for a single period (year).
x values are row names.
| Velliv_medium | Velliv_medium_long | Velliv_high | PFA_medium | PFA_high | mix_medium | mix_high | |
|---|---|---|---|---|---|---|---|
| 0 | 21.3 | 18.2 | 19.9 | 12.2 | 14.3 | 12.7 | 13.0 |
| 5 | 12.5 | 9.6 | 12.8 | 6.0 | 8.6 | 6.2 | 4.2 |
| 10 | 7.4 | 5.4 | 8.3 | 3.3 | 5.3 | 3.3 | 0.9 |
| 25 | 1.8 | 1.3 | 2.5 | 0.9 | 1.4 | 0.7 | 0.0 |
| 50 | 0.2 | 0.2 | 0.4 | 0.2 | 0.2 | 0.1 | 0.0 |
| 90 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 99 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
Chance of min gains of x percent for a single period (year).
x values are row names.
| Velliv_medium | Velliv_medium_long | Velliv_high | PFA_medium | PFA_high | mix_medium | mix_high | |
|---|---|---|---|---|---|---|---|
| 0 | 78.7 | 81.8 | 80.1 | 87.8 | 85.7 | 87.3 | 87.0 |
| 5 | 63.8 | 64.9 | 69.2 | 71.5 | 75.8 | 71.4 | 69.9 |
| 10 | 41.0 | 36.2 | 53.3 | 32.7 | 59.6 | 35.6 | 46.1 |
| 25 | 0.0 | 0.3 | 0.0 | 0.1 | 0.0 | 0.0 | 1.2 |
| 50 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 100 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
MC risk percentiles: Risk of loss from first to last period.
_a is simulation from estimated distribution of returns of
mix.
_b is mix of simulations from estimated distribution of
returns from individual funds.
_m is medium.
_h is high.
| Velliv_m | Velliv_m_long | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_h_a | mix_m_b | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 5.16 | 3.18 | 3.86 | 2.07 | 0.79 | 0.97 | 0 | 0.33 | 0.15 |
| 5 | 4.38 | 2.79 | 3.50 | 1.91 | 0.65 | 0.87 | 0 | 0.24 | 0.13 |
| 10 | 3.87 | 2.44 | 3.20 | 1.65 | 0.61 | 0.73 | 0 | 0.18 | 0.12 |
| 25 | 2.62 | 1.60 | 2.15 | 1.30 | 0.42 | 0.43 | 0 | 0.09 | 0.07 |
| 50 | 1.11 | 0.72 | 0.99 | 0.62 | 0.19 | 0.22 | 0 | 0.04 | 0.01 |
| 90 | 0.08 | 0.13 | 0.14 | 0.11 | 0.04 | 0.02 | 0 | 0.00 | 0.00 |
| 99 | 0.00 | 0.02 | 0.04 | 0.02 | 0.01 | 0.01 | 0 | 0.00 | 0.00 |
MC gains percentiles: Chance of gains from first to last
period.
_a is simulation from estimated distribution of returns of
mix.
_b is mix of simulations from estimated distribution of
returns from individual funds.
| Velliv_m | Velliv_m_long | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_h_a | mix_m_b | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 94.84 | 96.82 | 96.14 | 97.93 | 99.21 | 99.03 | 100.00 | 99.67 | 99.85 |
| 5 | 94.24 | 96.25 | 95.77 | 97.63 | 99.10 | 98.90 | 100.00 | 99.60 | 99.84 |
| 10 | 93.54 | 95.78 | 95.37 | 97.45 | 99.04 | 98.78 | 100.00 | 99.53 | 99.80 |
| 25 | 90.97 | 94.02 | 93.79 | 96.57 | 98.62 | 97.98 | 100.00 | 99.19 | 99.70 |
| 50 | 85.58 | 89.93 | 91.12 | 94.54 | 97.60 | 96.26 | 99.98 | 97.71 | 99.40 |
| 100 | 71.77 | 78.01 | 83.82 | 87.56 | 94.75 | 89.49 | 99.69 | 90.46 | 97.44 |
| 200 | 39.90 | 44.79 | 64.07 | 58.34 | 85.78 | 59.77 | 93.12 | 48.74 | 86.31 |
| 300 | 15.77 | 17.51 | 44.93 | 21.85 | 71.26 | 22.11 | 71.97 | 11.33 | 65.34 |
| 400 | 4.79 | 4.56 | 29.03 | 3.92 | 55.59 | 3.58 | 44.50 | 1.22 | 40.59 |
| 500 | 1.28 | 1.07 | 17.26 | 0.39 | 39.63 | 0.20 | 23.12 | 0.11 | 21.15 |
| 1000 | 0.00 | 0.00 | 0.70 | 0.00 | 2.50 | 0.00 | 0.28 | 0.01 | 0.09 |
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
| Velliv_medium | Velliv_medium_long | Velliv_high | PFA_medium | PFA_high | mix_medium | mix_high | |
|---|---|---|---|---|---|---|---|
| m | 0.048 | 0.052 | 0.065 | 0.058 | 0.084 | 0.059 | 0.082 |
| s | 0.120 | 0.115 | 0.150 | 0.123 | 0.121 | 0.088 | 0.071 |
| nu | 3.304 | 2.706 | 3.144 | 2.265 | 3.185 | 2.773 | 89.863 |
| xi | 0.034 | 0.505 | 0.002 | 0.477 | 0.018 | 0.029 | 0.770 |
| R-squared | 0.993 | 0.978 | 0.991 | 0.991 | 0.964 | 0.890 | 0.961 |
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is
calculated as the share of paths finishing below index 100.
## Number of paths: 10000
| Velliv_m | Velliv_m_long | Velliv_h | PFA_m | PFA_h | mix_m_a | mix_m_b | mix_h_a | mix_h_b | |
|---|---|---|---|---|---|---|---|---|---|
| mc_m | 278.437 | 292.991 | 405.817 | 319.768 | 555.977 | 324.319 | 301.885 | 501.245 | 475.563 |
| mc_s | 122.559 | 116.159 | 217.397 | 104.054 | 244.760 | 97.388 | 82.893 | 158.271 | 163.327 |
| mc_min | 1.432 | 0.000 | 0.000 | 0.000 | 0.626 | 0.000 | 18.971 | 129.151 | 25.692 |
| mc_max | 960.495 | 1003.923 | 1483.003 | 999.664 | 1748.121 | 673.260 | 1997.325 | 1335.386 | 1204.922 |
| dao_prob_pct | 0.000 | 0.010 | 0.010 | 0.010 | 0.000 | 0.010 | 0.000 | 0.000 | 0.000 |
| losing_prob_pct | 5.160 | 3.180 | 3.860 | 2.070 | 0.790 | 0.970 | 0.330 | 0.000 | 0.150 |
Ranking:
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_prob_pct | ranking | losing_prob_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 555.977 | PFA_h | 82.893 | mix_m_b | 129.151 | mix_h_a | 1997.325 | mix_m_b | 0.00 | Velliv_m | 0.00 | mix_h_a |
| 501.245 | mix_h_a | 97.388 | mix_m_a | 25.692 | mix_h_b | 1748.121 | PFA_h | 0.00 | PFA_h | 0.15 | mix_h_b |
| 475.563 | mix_h_b | 104.054 | PFA_m | 18.971 | mix_m_b | 1483.003 | Velliv_h | 0.00 | mix_m_b | 0.33 | mix_m_b |
| 405.817 | Velliv_h | 116.159 | Velliv_m_long | 1.432 | Velliv_m | 1335.386 | mix_h_a | 0.00 | mix_h_a | 0.79 | PFA_h |
| 324.319 | mix_m_a | 122.559 | Velliv_m | 0.626 | PFA_h | 1204.922 | mix_h_b | 0.00 | mix_h_b | 0.97 | mix_m_a |
| 319.768 | PFA_m | 158.271 | mix_h_a | 0.000 | Velliv_m_long | 1003.923 | Velliv_m_long | 0.01 | Velliv_m_long | 2.07 | PFA_m |
| 301.885 | mix_m_b | 163.327 | mix_h_b | 0.000 | Velliv_h | 999.664 | PFA_m | 0.01 | Velliv_h | 3.18 | Velliv_m_long |
| 292.991 | Velliv_m_long | 217.397 | Velliv_h | 0.000 | PFA_m | 960.495 | Velliv_m | 0.01 | PFA_m | 3.86 | Velliv_h |
| 278.437 | Velliv_m | 244.760 | PFA_h | 0.000 | mix_m_a | 673.260 | mix_m_a | 0.01 | mix_m_a | 5.16 | Velliv_m |
\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?
\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
c{ x_t + y_t }{ x_{t-1} + y_{t-1}} \[ \](x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)$$
\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).
Definition: R = 1+r
## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.
Then,
## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.
Average of returns:
## 0.5 * (R_x + R_y) = 1
So here the value of the pf at t=1 should be unchanged from t=0:
## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300
But this is clearly not the case:
## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175
Therefore we should take returns of average, not average of returns!
Let’s take the average of log returns instead:
## 0.5 * (log(R_x) + log(R_y)) = -0.143841
We now get:
## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076
So taking the average of log returns doesn’t work either.
Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.
## m(data_x): 0.027182
## s(data_x): 0.4186434
## m(data_y): 9.355286
## s(data_y): 3.453421
##
## m(data_x + data_y): 4.691234
## s(data_x + data_y): 1.725588
m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.
| m_a | m_b | s_a | s_b |
|---|---|---|---|
| 94.184 | 93.787 | 7.893 | 7.847 |
| 93.971 | 94.041 | 7.776 | 7.904 |
| 93.715 | 93.716 | 7.879 | 8.076 |
| 93.682 | 94.041 | 7.704 | 7.661 |
| 93.840 | 93.319 | 7.775 | 7.979 |
| 93.618 | 94.168 | 8.049 | 7.707 |
| 93.877 | 93.763 | 7.777 | 7.614 |
| 93.573 | 93.890 | 7.850 | 7.732 |
| 93.656 | 94.022 | 7.724 | 7.806 |
| 93.901 | 93.801 | 7.736 | 7.505 |
## m_a m_b s_a s_b
## Min. :93.57 Min. :93.32 Min. :7.704 Min. :7.505
## 1st Qu.:93.66 1st Qu.:93.77 1st Qu.:7.746 1st Qu.:7.672
## Median :93.78 Median :93.85 Median :7.777 Median :7.769
## Mean :93.80 Mean :93.85 Mean :7.816 Mean :7.783
## 3rd Qu.:93.90 3rd Qu.:94.04 3rd Qu.:7.872 3rd Qu.:7.890
## Max. :94.18 Max. :94.17 Max. :8.049 Max. :8.076
_a and _b are very close to equal.
We attribute the differences to differences in estimating the
distributions in version a and b.
The final state is independent of the order of the preceding steps:
So does the order of the steps in the two processes matter, when mixing simulated returns?
The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.
Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]
var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618
Our distribution estimate is based on 13 observations. Is that enough
for a robust estimate? What if we suddenly hit a year like 2008? How
would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.
## m s
## Min. :0.05940 Min. :0.04151
## 1st Qu.:0.06477 1st Qu.:0.06170
## Median :0.06704 Median :0.06573
## Mean :0.06897 Mean :0.06776
## 3rd Qu.:0.07281 3rd Qu.:0.07030
## Max. :0.08353 Max. :0.09079